models for s&p500 dynamics

8/13/2019 Models for S&P500 Dynamics

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Models for S&P500 Dynamics: Evidence from Realized

Volatility, Daily Returns, and Option Prices

Peter Christoersen Kris Jacobs Karim MimouniDesautels Faculty of Management, McGill University

December 16, 2008

Abstract

Most recent empirical option valuation studies build on the ane square root (SQR) sto-chastic volatility model. The SQR model is a convenient choice, because it yields closed-formsolutions for option prices. However, relatively little is known about the resulting biases.We investigate alternatives to the SQR model, by comparing its empirical performance withthat of ve dierent but equally parsimonious stochastic volatility models. We provide em-pirical evidence from three dierent sources: realized volatilities, S&P500 returns, and anextensive panel of option data. The three sources of data we employ all point to the sameconclusion: the SQR model is severely misspecied. The best of the alternative volatilityspecications is a model with linear rather than square root diusion for variance, whichwe refer to as the VAR model. This model captures the stylized facts in realized volatilities,

it performs well in tting various samples of index returns, and it has the lowest optionimplied volatility mean squared error in- and out-of-sample. It ts the option data betterthan the SQR model in several dimensions: it improves the t of at-the-money options, itprovides a more realistic volatility term structure and implied volatility smirk.JEL Classication: G12

Keywords: Stochastic volatility; option valuation; particle ltering; skewness; kurtosis;mean reversion.

Christoersen and Jacobs are also aliated with CIRANO and CIREQ and want to thank FQRSC, IFM2

and SSHRC for nancial support. Mimouni was supported by IFM2 and FQRSC. We are grateful for helpfulcomments from Yacine At-Sahalia (the Editor), Torben Andersen, Mikhail Chernov, Jerome Detemple, Jin Duan,

Bjorn Eraker, Bob Goldstein, Jeremey Graveline, Mark Kamstra, Nour Meddahi, Marcel Rindisbacher, StephenTaylor, and two anonymous referees. Correspondence to: Peter Christoersen, Desautels Faculty of Management,McGill University, 1001 Sherbrooke Street West, Montreal, Quebec, Canada, H3A 1G5; Tel: (514) 398-2869; Fax:(514) 398-3876; E-mail: [email protected].

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1 Introduction

Following the nding that Black-Scholes (1973) model prices systematically dier from marketprices, the literature on option valuation has formulated a number of theoretical models designedto capture these empirical biases. One particularly popular modeling approach has attempted to

correct the Black-Scholes biases by modifying the assumption that volatility is constant acrossmaturity and moneyness. Estimates from returns data and options data indicate that returnvolatility is time-varying, and modeling volatility clustering leads to signicant improvements inthe performance of option pricing models. It has also been demonstrated that it is necessary tomodel a leverage eect. The leverage eect captures the negative correlation between returnsand volatility, and thus generates negative skewness in the distribution of the underlying assetreturn.1

The existing literature has almost exclusively modeled volatility clustering and the leverageeect within an ane or square root structure. In particular, the Heston (1993) model, whichaccounts for time-varying volatility and a leverage eect, has been implemented in a large numberof empirical studies. Henceforth, we will refer to this model as the SQR model. The SQR modelis convenient because it leads to (quasi) closed-form solutions for prices of European options. It isnot surprising that the choice of model is partly driven by convenience: the estimation of optionvaluation models using large cross-sections of option contracts is computationally burdensomebecause of the need to lter the latent stochastic volatility. As the solution to the Heston (1993)SQR model relies on univariate numerical integration, it is relatively easier to estimate thanmany other models that require Monte Carlo simulation to compute option prices.

It is well recognized that the SQR model cannot capture important stylized facts. In orderto address the limitations of the square root structure, the model is often combined with modelsof jumps in returns and/or volatility, or multiple volatility components are used. 2 However,relatively few studies analyze non-ane stochastic volatility models, and therefore much less is

known about the empirical biases that result from imposing the ane square root structure onthe stochastic volatility dynamic in the rst place. Notable exceptions that investigate optionvaluation using non-ane stochastic volatility models are At-Sahalia and Kimmel (2007), Jones(2003) and Benzoni (2002). The non-ane model in Benzoni (2002) does not improve on theperformance of the Heston (1993) model. Jones (2003) analyzes the more general constantelasticity of variance (CEV) model using a bivariate time series of returns and an at-the-moneyshort maturity option, and a number of his specication tests favor the non-ane CEV model overthe SQR model. At-Sahalia and Kimmel (2007) also estimate dierent models using joint timeseries of the underlying return and a short-dated at-the-money option price or implied volatility.They also nd that the SQR model is misspecied, but the nature of the misspecication andthe empirical evidence are dierent from Jones (2003).

1 The leverage eect was rst characterized in Black (1976). For empirical studies that emphasize the im-portance of volatility clustering and the leverage eect for option valuation see among others Benzoni (2002),Chernov and Ghysels (2000), Eraker (2004), Heston and Nandi (2000), and Pan (2002).

2 For empirical studies that implement the Heston (1993) model by itself or in combination with dierent typesof jump processes, see for example Andersen, Benzoni and Lund (2002), Bakshi, Cao and Chen (1997), Bates(1996, 2000), Broadie, Chernov and Johannes (2007), Benzoni (2002), Chernov and Ghysels (2000), Huang andWu (2004), Pan (2002), Eraker (2004) and Eraker, Johannes and Polson (2003). Bates (2000) also introduces asquare root model with two volatility components.

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A number of papers estimate the SQR and alternative stochastic volatility specications onreturns data. These studies nd strong evidence against the SQR specication, see for exampleChacko and Viceira (2003) and Chernov, Gallant, Ghysels and Tauchen (2003). Similarly, therealized volatility literature seems to strongly favor logarithmic, non-ane SV specications. Seefor example, Andersen, Bollerslev, Diebold and Ebens (2001). Yet state-of-the-art option pricing

papers, such as for example Broadie, Chernov and Johannes (2007), continue employing the SQRmodel as a building block.

This paper further investigates the empirical implications of adopting a square root stochasticvolatility model for option valuation. We compare the empirical performance of the Heston(1993) square root stochastic volatility model with that of ve alternatives. In contrast withAt-Sahalia and Kimmel (2007) and Jones (2003), who compare the SQR model with more richlyparameterized stochastic volatility models, we deliberately choose alternative specications withthe same number of parameters as the SQR model. This approach also diers from several otherstudies that enrich the SQR model by adding for example jumps to the basic model, which alsoincreases the number of model parameters. We deliberately ask a very dierent question: is the

SQR model the best possible stochastic volatility model to build upon? If not, what are theempirical deciencies? Part of our motivation for keeping the number of parameters constant isour desire to make the in-sample comparison between models as meaningful as possible. It iswell known that in-sample model comparisons favor relatively richly parameterized models, butthat these in-sample model rankings can easily be reversed in out-of-sample experiments.

We conduct an extensive empirical investigation on the six stochastic volatility specicationsusing several dierent data sources. First, we use realized volatilities to assess the properties ofthe SQR model and to guide us in the search for alternative specications. Second, we estimatethe model parameters using maximum likelihood on returns only. Third, we employ nonlinearleast squares on a time series of cross-sections of option data with extensive cross-sectionalvariation across moneyness and maturity. Fourth, we value options using realized volatility and

VIX data as proxies for spot volatility. Finally, we investigate the robustness of the optionvaluation results with respect to the ltering method used. All evidence points to the sameconclusion: the ane SQR model is severely misspecied, and it is signicantly outperformedby the simple non-ane models we consider. Overall, the best of the alternative volatilityspecications is a model we refer to as the VAR model, which is of the GARCH diusion type.This model captures the stylized facts in realized volatilities, and it provides the best t for twoof the three return samples. Most importantly, when estimating the models using options data,it has the lowest option price mean squared errors in- and out-of-sample.

Our estimation on option contracts uses a rich panel of data. It therefore critically diersfrom At-Sahalia and Kimmel (2007) and Jones (2003), who estimate model parameters using

a bivariate time series. Indeed, to the best of our knowledge our analysis of the SQR modeluses substantially larger cross-sections of option contracts than any other available study thatexplicitly solve the ltering problem. Existing studies either solve the ltering problem and usea small cross-section of option data, or use a large cross-section and do not explicitly solve theltering problem. For example, At-Sahalia and Kimmel (2007), Chernov and Ghysels (2000),Eraker (2004) and Jones (2003) explicitly solve the ltering problem using a relatively limitedcross-section of option data. Studies that investigate a wider cross-section are for example Bakshi,Cao and Chen (1997), who estimate one cross-section at a time, and Bates (2000) and Huang and

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Wu (2004), who estimate a separate volatility parameter for each cross-section. Our estimation ona rich option data set while explicitly solving the ltering problem is a non-trivial contribution:One can arguably only reliably estimate the parameters that determine the model smirk andvolatility term structure when using such a wide range of option contracts.

We are able to estimate a large number of stochastic volatility models using comprehen-

sive cross-sections of options data thanks to our use of the particle lter. This methodologyis extensively used in the engineering literature, and has also recently been used for nancialapplications.3 Particle ltering provides a convenient tool for the analysis of latent factor modelssuch as stochastic volatility models. It is easy to implement, and it can be adapted to providethe best possible t to any objective function and any model of interest. In our opinion, thetrade-o with other empirical methods is a very favorable one in the context of computationallyintensive option valuation problems.

The paper proceeds as follows. In Section 2 we discuss the benchmark Heston (1993) model inlight of the evidence from realized volatilities. We also use the realized volatilities to help guidethe search for alternative specications. Section 3 discusses particle lter based estimation on

index returns and index options. Section 4 presents the empirical results obtained using returnand options data. Section 5 concludes. Evidence on the nite sample properties of the particlelter based estimator and on Monte Carlo based option pricing are provided in the appendix.

2 Stochastic Volatility Specications

This section discusses alternatives to the square root volatility specication in Heston (1993),which has become the standard building block for more elaborate models in the option valuationliterature. Note that it is relatively straightforward to write down more heavily parameterizedmodels that outperform the square root model. Trivially, a more heavily parameterized model

will always outperform a simpler nested model in-sample. Moreover, Bates (2003) points outthat even in short-horizon out-of-sample experiments, the richer models may outperform nestedmodels because the smile or smirk patterns in option prices is very persistent. This nding ismore likely if the models are re-estimated frequently, say daily as in the classic study by Bakshi,Cao and Chen (1997) for example.

Thus, we deliberately conne attention to models that have the same number of parametersas Hestons SQR model, and we estimate on a long sample of data. The models we considercan be thought of as alternative building blocks for more elaborate models containing jumps inreturns and volatility, as well as multiple volatility factors. However, such extensions are beyondthe scope of this paper. In our paper, the objective is to establish a well-specied volatilitydynamic, which in our opinion ought to precede the development of more elaborate versions of

a possibly misspecied basic model.3 See Gordon, Salmond and Smith (1993) and Pitt and Shepherd (1999). See Johannes, Polson and Stroud

(2008) for an application to returns data.

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2.1 Realized Variance: Implications for the Square Root Model

The Heston (1993) square root (SQR) model assumes that the instantaneous change in variance,V, has the following dynamics

SQR:dV =( V)dt + p

V dw (1)

where denotes the speed of mean reversion, is the unconditional variance, and deter-mines the variance of variance. The variance innovation dw is a Brownian motion, and we havecorr(dz;dw) =, where dz is the innovation in the underlying spot price, S, given by

dS= Sdt +p

VSdz (2)

where is the mean of the instantaneous rate of return. We will use the return specication (2)in each of the volatility models considered below.

In order to explore the implications of the SQR specication, consider the instantaneous

volatility dynamic implied by the model. Using Itos lemma, we can write

SQR: dp

V =(V)dt +1

2dw (3)

where the volatility drift (V) is a function of the variance level. Note that the SQR modelimplies that the instantaneous change in volatility should be Gaussian and homoskedastic: Vdoes not show up in any way in the diusion term for d

pV. This is a strong implication, which

can be quite easily evaluated empirically.As a rst piece of empirical evidence, we construct daily realized variances, RVt, from intraday

returns. We obtain intraday S&P500 quotes for 1996 through 2004 from which we construct agrid of two-minute returns. From these two-minute returns we construct robust realized variances

using the two-scale estimator from Zhang, Mykland, and At-Sahalia (2005).4 The two-scaleestimator is dened as

RVTSt =RVAvrt 0:0614RVAllt

where RVAvrt is the average of all the possible low-frequency RV estimates that use (in ourcase) 30-minute squared returns on the two-minute grid, and where RVAllt is the high-frequencyestimator summing over all the available two-minute squared returns.5 The coecient on RVAlltdepends on the frequency of the sparse estimators (30 minutes here) and of the high-frequencyestimator (2 minutes here). Zhang, Mykland, and At-Sahalia (2005) show that even in thepresence of market microstructure noise this estimator converges to the true integrated variancedened by Rtt1 Vd.

Consider now the left column of Figure 1. Using daily realized volatilities from 1996 through2004, the top-left panel of Figure 1 shows a quantile-quantile (QQ) plot of the daily realizedvolatility changes compared with the Gaussian distribution. The deviation of the data pointsfrom the straight line indicates that the Gaussian distribution is not a good assumption for dailychanges in volatility. The observed tails (both left and right) are considerably fatter than the

4 We scale up the open-to-close realized variance estimates to match the overall variance in close-to-close dailyreturns for the sample period.

5 See At-Sahalia, Mykland and Zhang (2005) for a discussion of the optimal sampling frequency.

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normal distribution would suggest. Now consider the middle-left panel in Figure 1, which scatterplots the daily volatility changes against the daily volatility level. According to the SQR model,this scatter plot should not display any systematic patterns. However, as the volatility levelincreases on the horizontal axis, a cone-shaped pattern in the daily volatility changes on thevertical axis is apparent. The bottom-left panel of Figure 1 conrms this nding: it scatter plots

the absolute daily volatility changes against the daily volatility level. A simple OLS regressionline is shown for reference. Notice the apparent positive relationship between the volatility leveland the magnitude of the volatility changes. This pattern is in conict with the homoskedasticvolatility implication of the SQR model.

Using Itos lemma, the dynamics for log variance in the SQR model can be written as

SQR: d ln(V) =(V)dt + 1p

Vdw (4)

The right column of Figure 1 summarizes the empirics of the logarithms of the realized vari-ances. The top panel shows that daily changes in the realized log variances follow the Gaussian

distribution quite closely. The result clearly diers from the corresponding top left panel forthe daily volatility changes. Furthermore, the scatter of daily log variance changes against thelog variance level in the middle-right panel of Figure 1 does not reveal a cone-shaped pattern.The bottom-right panel in Figure 1 conrms this result: it shows a virtually horizontal linewhen regressing absolute changes in log variance on log variance levels. The absence of a clearrelationship between changes in the log variances and the log variance level in Figure 1 castsfurther doubt on the SQR specication, for which the instantaneous changes in log variances areheteroskedastic, as is evident from (4).

2.2 Alternative Specications for Variance Dynamics

The evidence in Figure 1 suggests specifying the variance dynamic as follows

VAR:dV =( V)dt + V dw (5)

which has the property that ln(V)is homoskedastic.6

d ln(V) =(V)dt + dw

Heston (1997) and Lewis (2000) suggest another alternative to the SQR model: the so-calledthree halves model dened by

3/2N:dV =V( V)dt + V3=2dw (6)In addition to a dierent power on Vin the diusion term, the 3/2N model has the interestingimplication that the variance drift is nonlinear in the level of the variance. We highlight thisfeature by denoting the model 3/2N. This allows for a potentially fast reduction in volatility

6 Although it would seem natural from the realized variance analysis above, we do not consider the homoskedas-tic log SV model where d ln(V) = (ln(V)) + dw, because it typically performs very similar tobut slightlyworse thanthe VAR model in (5).

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following unusually large volatility spikes, such as October 1987, and for slow increases in volatil-ity during prolonged low-volatility episodes, such as the mid 1990s. The 3/2N model also leadsto a closed-form solution for European call option prices (see Lewis (2000)), even though to thebest of our knowledge the model has not yet been implemented using this solution.

So far we have considered three dierent specications of the diusion and two dierent

specications for the drift ofV. We can think of the three models considered so far as belongingto the class

dV =Va( V)dt + Vbdw, fora =f0; 1gandb =f1=2; 1; 3=2g (7)Although we will focus our discussion on the SQR, VAR and 3/2N models, this frameworkencompasses a total of six models, which we denote

a b Name0 1/2 SQR1 1/2 SQRN0 1 VAR1 1 VARN0 3/2 3/21 3/2 3/2N

All of these models will be estimated below.The literature contains several good discussions of the properties of stochastic dierential

equations. See for instance Karlin and Taylor (1981). At-Sahalia (1996) contains a discussion ofthe properties of general interest rate processes and Lewis (2000) contains a treatment of certainvariance processes. Jones (2003) provides an excellent discussion of the CEV stochastic varianceprocess. He shows that for a model with a = 0and b >1, zero and plus innity are inaccessiblevalues for the stochastic variance. He also shows that a solution to the variance SDE exists and

is unique, and that the stationary distribution exists. Thus Jones (2003) covers the model werefer to as 3/2. Similar arguments can be used to show that these results hold also for the VARmodel, the VARN model and the 3/2N model. For the SQRN model, all but the existence of thestationary distribution can be shown using similar arguments as well.

2.3 Volatility Risk Premia

In order to compute option prices, which depend on the price of volatility risk, we need to specifya volatility risk premium relationship for each of the above six models. Heston (1993) assumesthat the volatility risk premium (S;V;t) is equal to V, so that the risk neutral dynamic for

the SQR model isSQR: dS= rSdt +

pV S dz (8)

dV = (( V) + V) dt + p

V dw (9)

= ( )(=( ) V)dt + p

V dw

wheredz anddw are Brownian motions under the risk-neutral measure, with corr(dz; dw) =: Notice that the variance dynamic takes the same form under the physical and risk neutralmeasures.

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To provide some intuition for the properties of risk-neutral volatility, Figure 2 plots variousstylized facts for the model-free, option-implied VIX volatility index, provided by the CBOE.The VIX is measured in annualized percentage standard deviations. Similar to Figure 1, wereport the QQ plot of the VIX in the top left panel. The daily VIX changes are scattered againstthe VIX level in the center-left panel, and the daily absolute VIX changes are scattered against

the VIX level in the bottom-left panel. In the right column, we provide the same analysis usingthe natural logarithm of the VIX rather than the VIX level.7 The similarities with the RV-basedplots in Figure 1 are quite striking. The level of the VIX displays strong evidence of nonnormalityand heteroskedasticity. The distribution of the logarithm of the VIX appears to be much closerto the normal distribution, and shows much less evidence of heteroskedasticity.

A key lesson from Figure 2 is that the (model-free) risk-neutral volatility distribution hassimilar features to the (model-free) physical volatility distribution in Figure 1. Therefore, whendeciding on the risk premium specication for the non-ane models, we choose a form thatpreserves the functional form of the spot variance process under the two measures. In particular,we use the following functional form for the variance risk premium

(S;V;t) =Va+1 (10)This specication ensures that the variance dynamic takes the same form under the two measures,as is the case in most available empirical studies that estimate the Heston (1993) SQR model.Under the risk neutral measure, we have

dS=rSdt +p

V Sz (11)

dV = ( )Va(=( ) V)dt + Vbdwwithcorr(dz; dw) =:

In all six models we investigate, the risk-neutralization can be obtained using a no-arbitrageargument. A utility-based argument behind the risk neutralization can be explicitly characterized

for some cases when assuming log-utility. See Lewis (2000) for a thorough discussion of theseissues.

Note that the risk-neutral process in (11) can be rewritten

dS=rSdt +p

V Sz (12)

dV =Va( V)dt + Vbdwwhere = (), and ==. This representation highlights that cannot be identiedin any of the models when using only options in estimation. Below we perform two types ofvaluation exercises: one with as well as the other parameters estimated using option data andreturns jointly, and one with equal to zero and the other parameters estimated from returnsonly.

2.4 Computing Option Prices

Heston (1993) demonstrates that the SQR model admits a closed form solution for option prices,which can be written as

C(Vt) =StP1;t Xer(Tt)P2;t (13)7 Note that the analysis for the natural logarithm of the VIX is equivalent to the analysis of the natural

logarithm of the squared VIX.

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where P1;t andP2;t are computed using numerical integration of the conditional characteristicfunction.

Heston (1997) and Lewis (2000) show that for the 3/2N model, a similar closed-form solutionis available, but in this case the characteristic function involves the gamma and conuent hyper-geometric function with complex arguments. To the best of our knowledge, this solution is very

numerically intensive and has not yet been implemented empirically. Barone-Adesi, Rasmussenand Ravanelli (2003) and Sabanis (2003) develop approximate analytical solutions for the VARmodel, but as far as we know an exact solution is not available.

In order to ensure that dierences across models are not driven by a particular numericaltechnique, we compute all model option prices using Monte Carlo. Thus the call option pricesare computed via the Monte Carlo sample analogue of the discounted expectation

C(Vt) =er(Tt)Et [ST X; 0] (14)

where the expectation is calculated using the risk neutral measure.In order to calculate Monte Carlo prices, we use 1,000 simulated paths and a number of

numerical techniques to increase numerical eciency, namely the empirical martingale methodof Duan and Simonato (1998), stratied random numbers, antithetic variates and a Black-Scholescontrol variate technique. In the appendix we use the SQR model to demonstrate that the MonteCarlo prices match closely the prices computed via numerical integration.

3 Estimation Using the Particle Filter

The SQR model has been investigated empirically in a large number of studies. It is often usedas a building block, together with models of jumps in return and volatility. For our purpose,it is important to note that when estimating the model on option data, a number of dierenttechniques can be used. First, the models parameters can be estimated using a single cross-section of option prices treating Vt as a parameter. This is done for example in Bakshi, Cao andChen (1997). A second type of implementation of the SQR model uses multiple cross-sections ofoption prices but does not use the information in the underlying asset returns. Instead, for everycross-section a dierent initial volatility is estimated, leading to a highly parameterized problem.This approach is taken for instance in Bates (2000) and Huang and Wu (2004). A third groupof papers provide a likelihood-based analysis of the stochastic volatility model. See for exampleAt-Sahalia and Kimmel (2007), Bates (2004), Jones (2003), and Eraker (2004), who provides aMarkov Chain Monte Carlo analysis. A likelihood-based approach can combine information fromthe option data and the underlying returns, and impose consistency between the physical and

risk neutral dynamics in estimation. Both the return data and the option data carry a certainweight in the objective function. Finally, Chernov and Ghysels (2000) use the ecient methodof moments and Pan (2002) uses a method of moments technique as well. These methods canalso combine information in option data with the information in underlying returns.

The empirical challenge in stochastic volatility models is that the spot volatility Vt is anunobserved latent factor. In order to estimate the parameters in stochastic volatility models,we thus need to apply a ltering technique using observed index returns. The last two groupsof papers discussed above explicitly solve the ltering problem. The ltering problem is also

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explicitly considered in some papers that use return data to estimate continuous-time stochasticvolatility models, such as Chernov, Gallant, Ghysels and Tauchen (2003).

We provide an alternative implementation of the ltering problem for the six stochasticvolatility models under investigation using the particle lter (PF) algorithm. As shown byGordon, Salmond and Smith (1993), the PF oers a convenient lter for nonlinear models such as

the stochastic volatility models we consider here. The PF has recently been applied by Johannes,Polson and Stround (2008), who use return data to estimate continuous-time stochastic volatilitymodels with jumps. But because the PF procedure is relatively new in nance, we discuss theimplementation of this method in some detail.

We use the PF both for maximum likelihood estimation on returns and for nonlinear leastsquares estimation on option prices. Below, we rst describe the PF, then the maximum likeli-hood importance sampling (MLIS) estimation methodology we use to estimate the models usingreturn data, and nally the nonlinear least squares importance resampling (NLSIS) estimationmethodology we use to estimate the models using option data.

3.1 Volatility Discretization

Working with log returns, we can write the generic SV process as

d ln(S) =

1

2V

dt +

pV dz (15)

dV = Va( V)dt + Vbdw (16)Note that the equations in (15) and (16) specify how the unobserved state is linked to observedstock prices. This relationship allows us to infer the volatility path using the returns data. Werst need to discretize (15)-(16). There are dierent discretization methods and every schemehas certain advantages and drawbacks. We use the Euler scheme, which is easy to implement

and has been found to work well for this type of applications. 8 Discretizing (15)-(16) gives

ln(St+1) = ln(St) +

1

2Vt

+p

Vtzt+1 (17)

Vt+1 = Vt+ Vat ( Vt) + Vbt wt+1 (18)

We implement the discretized model in (17) and (18) using daily returns, but all parameters willbe expressed in annual units below. We now describe the volatility ltering procedure.

3.2 Volatility Filtering

The particle lter algorithm relies on the approximation of the true density of the state Vt+1by a set ofN discrete points or particles that are updated iteratively through equations (17)

and (18). The lter is implemented using particles or draws,

Vjt+1Nj=1

; from the empirical

distribution ofVt+1, conditional on particles or draws,

VjtNj=1

;from the empirical distribution of

Vt. Our particular implementation of the particle lter is referred to as the Sampling-Importance-Resampling (SIR) particle lter and follows Pitt (2002).

We recursively proceed through the following three steps for t = 1;:::;T1:8 See for example Eraker (2001).

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3.2.1 Step 1: Simulating the State Forward: Sampling

We rst simulate the state forward by computing N raw particles { ~Vjt+1}Nj=1 from the set of

smooth resampled particles {Vjt }Nj=1 using equation (18), taking the correlation between returns

and variance into account. We x the particles in the rst period as Vj1 =, for allj .9

We can then get correlated shocks via

zjt+1 =

ln(St+1)ln(St)

1

2Vjt

=

qVjt (19)

wjt+1 = zjt+1+

p1 2"jt+1

where"jt+1 are independent random draws from the standard normal so that corr(zjt+1; "

jt+1) = 0

andcorr(zjt+1; wjt+1) =. Substituting (19) into (18) we get

~Vjt+1= Vjt + V

jt

a

Vjt + V

jt

bwjt+1 (20)

This simulatesNraw particles and thus provides a set of possible values ofVt+1.

3.2.2 Step 2: Computing and Normalizing the Weights

At this point, we have a vector ofNpossible values ofVt+1 and we know according to equation(17)that given the other available information, Vt+1 is sucient to generateln(St+2):Therefore,equation(17) oers a simple way to evaluate the likelihood that the observation St+2 has beengenerated by Vt+1: Hence, we are able to compute the weight given to each particle (or thelikelihood or probability that the particle has generated St+2). The weight is computed asfollows

~Wjt+1= 1q

2 ~Vjt+1

exp0B@1

2

lnSt+2

St+1 12 ~Vjt+1

2

~Vjt+1

1CA (21)forj = 1;::;N:Finally, we normalize the weights via

Wjt+1=~Wjt+1PN

j=1~Wjt+1

: (22)

3.2.3 Step 3: Smooth Resampling

The setn

Wj

t+1oNj=1 can be viewed as a discrete probability distribution of ~Vt+1 from which we

can resample. Let the ordered particles be dened by ~V(j)t+1 then the corresponding cumulated

sum of ordered weights, W(j)t+1; gives the discrete, step-function CDF corresponding to the dis-

tribution ofVt+1. Pitt (2002) proposes smoothing the W(j)t+1-based CDF in order to ensure that

the subsequent likelihood function is smooth in the model parameters, which renders numerical

9 In the returns-based MLIS estimation, t= 1 is simply the rst day of observed returns. In the options-basedNLSIS estimation, t= 1 is one year prior to the rst available option quote.

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optimization feasible. The smooth bootstrap in Efron and Tibshirani (1993) oers a possiblebut computationally expensive solution to this problem. Fortunately, Pitt (2002) provides avery ecient smoothing approach which is crucial for our extremely intensive option valuationapplication.

Pitts (2002) approach is simple and intuitive: The discrete, step-function empirical CDF

implied by W(j)t+1can be converted to a smooth continuous CDF by partitioning the sample spacefor Vt+1 into regions delimited by the observed ordered particles. Let region j be dened by

Bj = [ ~V(j)t+1;~V

(j+1)t+1 ]: Now dene the probability ofVt+1 being in region j as

Pr (Vt+12Bj) = 12

W(j)t+1+ W

(j+1)t+1

; j = 2; 3;:::N2 (23)

and dene the conditional density for regioni by

g (Vt+1jVt+12Bj) = 1

~V(j+1)t+1 ~V

(j)t+1

; j = 2; 3;:::N2 (24)

By appropriately dening the probability distribution in the end-regions, that is for j = 1and j = N 1, we ensure that the probabilities sum to one and that the continuous CDFcorresponding to(24)passes through a midpoint of each step in the discrete CDF. As N becomeslarge, the continuous resampled CDF will converge to the discrete CDF, which in turn willconverge to the true CDF. We refer to Pitt (2002) and Smith and Gelfand (1992) for the details.

Note from (24) that we are assuming a uniform distribution on each region of the samplespace. Because the uniform distribution is easy to invert, this is a computationally ecient wayto get a set of smooth particles. We resample the particles a total ofNtimes in proportion to

their smooth likelihood in(23)and (24)to get the smooth resampled particles, Vjt+1

N

j=1.

The ltering for period t+ 1 is now done. The ltering for period t+ 2 starts in step 1above by computing shocks for period t+ 2 in (19) using the smooth resampled particles, Vjt+1,as inputs. Repeating steps 1-3 for allt yields a discrete distribution of ltered spot variances oneach day.

3.3 Maximum Likelihood Estimation

Our rst empirical strategy uses a long sample of daily S&P500 index returns to estimate each ofthe SV models by maximizing the likelihood. To this end, we need an estimation methodologywhich allows for models with a latent volatility factor.

Pitt (2002) builds on Gordon, Salmond and Smith (1993) to show that the parameters of latent

factor models in general, and of the SV model in (17) and (18) in particular, can be estimatedby maximizing the Maximum Likelihood Importance Sampling criterion, which is simply denedby

MLIS(;;;;) =TXt=1

ln

1

N

NXj=1

Wjt

! (25)

As described in the previous section, the particle weights, Wjt, are determined via the condi-tional likelihood of particle j at time t. These individual likelihoods in turn are given from the

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model specication in (17) and (18), taking into account that zt andwt are correlated normalrandom variables. The MLIS criterion then simply averages the particle weights across particles,takes logs, and sums over time to create a log likelihood function.

A key challenge in the use of the MLIS for estimation and inference is that it is not generallysmooth in the underlying parameters. However, as discussed above, Pitts (2002) ingenious

implementation of the particle lter, where the resampling in Step 3 is done in a smooth fashion,ensures that the MLIS criterion is smooth in the parameters. This smoothing drastically improvesthe numerical optimization performance and it enables us to compute reliable parameter standarderrors using conventional rst-order techniques.

In the appendix, we provide a small scale Monte Carlo experiment comparing the MLISestimator to other estimators available in the literature. We nd that the MLIS estimatorperforms well in comparison with much more computationally intensive methods.

3.4 Nonlinear Least Squares Estimation

Our second empirical strategy is to take a large panel of options traded on the S&P500 indexand estimate each of the SV models by minimizing the option pricing errors on this sample. Forall the SV models, our implementation uses the nonlinear least squares importance sampling(NLSIS) estimation technique, which minimizes the following mean squared implied volatilityerror

I V M S E (;;;;;) = 1

NT

Xt;i

IVi;t BS1i

Ci( Vt)

2(26)

where I Vi;t is the Black-Scholes implied volatility corresponding to the market price of option iquoted on dayt. Ci

Vt

is the model price evaluated at the ltered variance, and BS1

Ci( Vt)

denotes the Black-Scholes inversion of the model option price.10 The total number of options

in the sample is denoted by NT =

TPt=1 N

t, where T is the total number of days included in theoptions sample, and where Nt is the number of options with various strike prices and maturitiesincluded in the sample at datet. The ltered spot variance Vtis simply the average of the smoothresampled particles

Vt = 1

N

NXj=1

Vjt (27)

Our implementation minimizing (26) is dierent from existing studies that estimate SV mod-els from option data while solving the ltering problem. Most other studies are likelihood-based.One advantage of our approach is that it is relatively straightforward and fast, which allows

us to estimate the models using much more extensive cross-sections of options. In our opinion,estimation using (26) has an additional advantage, because matching the objective function usedin parameter estimation with the function subsequently used to evaluate the models ensuresthe best possible performance of the models in- and out-of-sample. This is motivated by theinsights of Granger (1969), who demonstrates that the choice of objective function (also labeled

10 Notice that we could alternatively compute the model price as the weighted average of the option pricescomputed for each particle. However, such an approach would be computationally very costly. Note that if thedistribution of particles is centered around the mean, the two approaches will yield very similar results.

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loss function) is an integral part of model specication. It follows that estimating a model usingone objective function and evaluating it using another one amounts to a suboptimal choice ofobjective function. Christoersen and Jacobs (2004) demonstrate that this issue is empiricallyrelevant for the estimation of the deterministic volatility functions in Dumas, Fleming and Wha-ley (1998). We thus choose to implement the SV models in a way that is consistent with these

insights. Notice however that our use of the particle ltering algorithm is completely general:we can apply this technique to any volatility model and using any well-behaved loss functioninvolving option prices and underlying returns, including a likelihood function.

Our optimization algorithm minimizes (26) using an iterative procedure on the structuralparameters. At each iteration, the volatility is ltered through time using the current parametervalues and the information embedded in observed returns. Using the ltered volatility and thestructural parameters, option prices are computed and the IVMSE is calculated. The proceduresearches in the structural parameter space until an optimum is reached.

4 Empirical Results

This section presents our empirical results obtained using returns and options data. First, wediscuss the return-based MLIS estimation results and dierences in volatility sample paths. Sec-ond, we introduce the option data set and use the MLIS parameters to compare option valuationperformance across models. Third, we estimate the models using NLSIS estimation and ana-lyze the patterns in the option valuation errors. Fourth, we check the robustness of our resultsto the use of dierent volatility proxies and dierent volatility ltering techniques. In all ourimplementations of the PF, we use ve hundred particles.

Out-of-sample experiments, in which the models are evaluated using data not used in estima-tion, are conducted in two dierent ways. First, we use purely return-based MLIS estimates to

value options. Second, we use NLSIS-based parameters obtained from Wednesday call optionsto value Wednesday put options and Thursday call options. We use the same calendar periodfor the out-of-sample study so as to avoid the impact of structural breaks in volatility. Becausewe are comparing equally parsimonious models estimated on a large sample, the Bates (2003)critique mentioned in Section 2, that more heavily parameterized models are favored in suchout-of-sample experiments, does not apply.

4.1 MLIS Estimation Results from Index Returns

As mentioned above, our rst empirical strategy uses a long sample of daily S&P500 index returnsand estimates each of the SV models by maximizing the model t for this sample. First, recall

the SV model specications we consider

dV =Va( V)dt + Vbdw, fora =f0; 1gandb =f1=2; 1; 3=2g:

We use daily S&P500 returns from CRSP and estimate the physical parameters by maximizing

MLIS(;;;) =TXt=1

ln

1

N

NXj=1

Wjt

!

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In the MLIS optimization, the return drift is xed at the sample average daily return inall models. The optimal parameters as well as the MLIS optimum values and the volatilityproperties for each model are given in Table 1. Consider rst the MLIS objective values. Wereport three sets of values. First, we present results for 1996-2004, which matches the sampleperiod we will use in the subsequent option estimation analysis. Second, we present results for

1989-2004, which is a longer sample period that does not include the 1987 crash, and nallyresults for 1985-2004, which includes the crash. Note rst that the ranking of models is verystable across the various samples. The VAR(a= 0; b= 1) model or the 3/2N(a= 1; b= 3=2)model is always best or second best. The SQR(a= 0; b= 1=2) model is ranked fourth or fthand the SQRN specication (a = 1; b = 1=2) is always worst. While we do not have inferenceprocedures for these MLIS objective values for non-nested models, recall that in standard LRtests, adding one parameter to a model is signicant at the 5% level if the log-likelihood increasesby approximately two points. The dierences in objective values across these models that havethe same number of parameters therefore appear to be quite large.

Table 1 also contains the parameter estimates for the 1996-2004 sample period and the right-

most four columns of Table 1 provide the rst four sample moments of the ltered volatility. Notethat the mean ltered volatility is similar across models. Note also that the and parametersare not directly comparable between linear (a = 0) and nonlinear (a = 1) drift specications.Figure 3 therefore plots the drift function for all models (solid lines) as a function of the levelof variance. The linear drift specications are given in the left column and the nonlinear driftsin the right column. The square root diusions are in the top row, the linear diusions in themiddle row, and the 3/2N diusions in the bottom row. The dierences between linear andnonlinear drift specications are most evident for large values of the spot variance where themean-reversion in the nonlinear models is much stronger.

The diusion parameter is comparable within a given diusion specication (i.e. for agiven value ofb), but not across diusion specications. In order to facilitate comparisons of the

models, Figure 3 shows the diusion functions (dashed lines) plotted against the value of the spotvarianceV. Notice that in comparison with the VAR diusion specication (b= 1)in the middlerow, and the 3/2 specications(b= 3=2) in the bottom row, the SQR and SQRN specications(b = 1=2) in the top row enable much less diusion in the variance when the variance level islarge.

The nal parameter estimate is that of, which captures the correlation between the shocksto return and variance. Ranging from 0.7876 to 0.7411, the estimate of is stable acrossmodels and relatively large in magnitude. However, recent studies, including Jones (2003) andAt-Sahalia and Kimmel (2007), have obtained similarly large correlation estimates.

4.2 Properties of the Filtered Spot VolatilitiesWe now present some more empirical evidence on the dierences in model performance overtime. We rst plot the path of ltered volatility over time in Figure 4. Subsequently, we plotthe path of the conditional volatility of variance in Figure 5. Finally, in Figures 6-8 we assessthe distributional characteristics of the ltered spot volatilities for the three models with lineardrift, and compare them to the properties of the realized volatilities in Figure 1.

Figure 4 plots the ltered volatility pathsp

Vt for each model during the 1996 to 2004 sample

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period. All volatility paths are shown on the same scale, going from zero to 70 percent volatilityin annual terms. Naturally, the overall pattern in volatility over time is similar across models.However, when volatility increases, it tends to do so much more sharply in the 3/2 modelsand somewhat more sharply in the linear (VAR) diusion models when compared with thesquare root diusion models. The VAR and 3/2 diusions thus exhibit more spikes in volatility

when compared with the SQR model. Examples of this include September 1998 (LTCM/Russiadefault), September 2001 (9/11), and July 2002.

The path of spot variances in Figure 4 is of course crucial for the models option valuationperformance over time. However, the time paths of the higher moments are equally important.We can dene the model-based conditional variance of variance as

V art(Vt+1) =2 V2bt : (28)

Figure 5 shows the path of the annualized conditional volatility of variance for each model, i.e.the square root of the expression in (28). We again use the same scale for all the plots soas to emphasize the dierences across models. Notice how high-volatility episodes such as theSeptember 1998 LTCM debacle and Russia default, as well as the July 2002 stock market decline,lead to pronounced dierences between the benchmark models, namely the SQR model in thetop-left panel, the VAR model in the middle-left panel, and the 3/2N model in the bottom-rightpanel.

In Figures 6 through 8, we report on the distributional properties of the levels of the spotvolatilities (left column) and the natural logarithms of the variances (right column), for the SQR,VAR, and 3/2 models with linear drifts. As in Figures 1 and 2, we report the QQ plots in the toprow, the daily changes scattered against the volatility (or log variance) in the middle row, andthe daily absolute changes scattered against volatility (or log variance) in the bottom row. Thedierences across the three models are quite striking. The QQ plots reveal that the SQR model

in Figure 6 has close to normally distributed spot volatilities but nonnormal log spot variances.The VAR model in Figure 7 has non-normal volatilities but approximately normal log variances,and the 3/2 model in Figure 8 has strong non-normality in both the spot volatilities and the logvariances. The scatter plots are also quite revealing. Notice that the SQR model in Figure 6 hasclose to homoskedastic volatility changes but heteroskedastic log variance changes. The VARmodel in Figure 7 has heteroskedastic volatility changes but close to homoskedastic log variancechanges. The 3/2 model in Figure 8 is characterized by heteroskedastic changes in both volatilityand log variance.

Properties of the distribution of annualized ltered volatilities can also be inferred from the

standard deviation, skewness, and kurtosis of the

pVt paths provided in Table 1 for all six

models. Not only is the standard deviation of volatility dierent across models, the VAR, 3/2

and 3/2N models are also characterized by much larger skewness and excess kurtosis of volatilitycompared to the SQR model. This of course conrms the ndings in Figures 6-8.

Finally, note that when comparing the model-based spot volatilities and log variances inFigures 6-8 to the model-free realized volatilities and log variances in Figure 1, it appears thatthe VAR model seems to best capture the distribution of volatility.

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4.3 Option Data

We conduct our empirical option-based analysis using S&P500 index option data for the 1996-2004 period. We only use Wednesday and Thursday options data. For the NLSIS based in-sampleanalysis, we use Wednesday call option data. Wednesday is the day of the week least likely to

be a holiday. It is also less likely than other days such as Monday and Friday to be aected byday-of-the-week eects. The decision to pick one day every week is to some extent motivatedby computational constraints. The optimization problems are fairly time-intensive, and limitingthe number of options reduces the computational burden. Using only Wednesday data allows usto study a long time-series, which is useful considering the highly persistent volatility processes.An additional motivation for only using Wednesday data is that following the work of Dumas,Fleming and Whaley (1998), several studies, including Heston and Nandi (2000), have used thissetup. We use the parameters obtained in the NLSIS estimation exercise together with eitherWednesday put options or Thursday call options in order to assess the out-of-sample performanceof the models.

Panel A of Table 2 presents descriptive statistics for the options data for the 1996-2004

Wednesday closing call option data by moneyness and maturity. The sample includes a total of16,506 Wednesday call option contracts with an average mid-price of $46.05 and average impliedvolatility of 20.26%. The implied volatility is largest for the in-the-money options reecting thewell-known volatility smirk in index options. The average implied volatility term structure isroughly at during the period. The average bid-ask spread is $1.60 corresponding to about 3.5%of the average mid-price of $46.05.

Panel B of Table 2 provides the option data characteristics for put options recorded at theclose of each Wednesday. We observe a total of 23,035 puts with an average mid-price of $40.87and an average implied volatility of 21.80%. The average bid-ask spread is $1.53 or about 3.7%of the average mid-price. The implied volatility smirk is evident in the put options as well: The

out-of-the-money put options have an average of 24.26% implied volatility compared with around20% for at-the-money puts. Panel C of Table 2 shows that the Thursday call option sample has15,390 options with characteristics that are similar to those of the Wednesday call option data.

4.4 Option Valuation with MLIS Estimates

This section presents results from option valuation using the MLIS estimates in Table 1, obtainedusing returns only. In this case, we cannot estimate the volatility risk premium . We thereforesetto zero, implying that the variance dynamics are the same under the two measures. Clearly,setting to zero will worsen the t of all the models. Consequently, the focus in this analysis isnot primarily on the level of the pricing errors, but rather on the relative t across models. In

the subsequent section, we will estimate along with the other parameters, using both returnand option data.

Table 3 contains the results for the MLIS-based option valuation exercise. We report the tof the dierent SV models using our three sets of options data (Wednesday calls, Wednesdayputs, and Thursday calls) and the following three criteria: implied volatility root mean squarederror, IV RMSE; dollar root mean squared error, $RMSE, which uses the option mid-prices;and out-of-spread root mean squared error,ORMSE, which sets the dollar pricing errors to zeroif the model price falls between the observed bid and ask price for a given option.

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Table 3 uses the ane SQR model as the benchmark model and it reports the RMSE ratiobetween each model and the SQR model. It also reports the Diebold and Mariano (1995) testof the null hypothesis that the weekly MSE of each model is equal to that of the SQR model.Boldface font indicates signicance at the 5% level.

Table 3 shows that for each of the 45 pairwise comparisons made, the RMSE ratio with the

SQR model is smaller than one. Thus the SQR model is outperformed by all other models for allthree criterion functions and for all three sets of option data. The ratios tend to be the lowestfor theOSRMSEloss function, suggesting that the dierences between the SQR and the othermodels are economically relevant. From the Diebold-Mariano test statistics we see that a largemajority of the dierences with the SQR model are statistically signicant. A comparison of theve non-ane models suggests that none of the models systematically outperforms the others.A given models performance depends on the criterion and the data used, but in general thedierences between the ve non-ane models are relatively small.

4.5 NLSIS Estimation on Option Prices and Returns

While the MLIS-based results in Table 3 are of interest, it is likely that a much better t can beobtained by estimating all the parameters including the volatility risk premium using informationfrom both underlying returns and option prices. This section presents such results usingNLSISestimation.

Table 4 contains the parameter values obtained from minimizing the option implied volatilitymean-squared-error, dened above as

I V M S E (;;;;) = 1

NT

Xt;i

IVi;t BS1i

Ci( Vt)

2(29)

In the NLSIS optimization, the return drift, , is xed at the sample average daily return in all

models, as was the case in MLISestimation.Consider rst the in-sample root mean squared error (IVRMSE) column in Table 4, which

uses Wednesday call options to compare the models. The SQR model performs the worst with anIVRMSE of 3.32%, and the VAR model performs the best with an IVRMSE of 2.86%. The RMSEthus diers by about 14% between the SQR and the VAR model. Note that this improvementin t is obtained without adding any parameters to the model. The 3/2N model is about 11%better than the SQR model and the VARN model with nonlinear drift and linear diusion(a = 1; b = 1) is about 10% better than the SQR model. The VAR, VARN and 3/2N modelssignicantly outperform the SQR model when judged by the Diebold-Mariano statistic. Theremaining two models do not signicantly outperform the SQR model.

Consider next the out-of-sample columns in Table 4, which use Wednesday puts and Thursdaycalls respectively. For Wednesday put options, the VAR model outperforms the SQR model byapproximately 19%, and the VAR model is the only model that signicantly outperforms theSQR model according to the Diebold-Mariano test. Using Thursday call options, the VAR modelagain performs the best and outperforms the SQR model by a signicant 17%. For Thursdaycalls, the 3/2N model improves upon the SQR model by 11%, which is also signicant when

judged by the Diebold-Mariano test. The VARN model has an IVRMSE that is 9% below theSQR, but this dierence is not statistically signicant. The remaining three models show minorand statistically insignicant improvements.

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The overall improvement in IVRMSE provided by the VAR model over the benchmark SQRmodel is quite substantial. To judge the magnitude of the improvement in t, the most naturalreference point is the rich literature that uses Poisson jumps in returns and/or volatility in con-

junction with an SQR stochastic volatility model. The evidence on the in-sample and especiallythe out-of-sample improvement provided by including jump processes is inconclusive, with some

studies nding moderate improvements, and others concluding that there is no improvement int.11 The improvement in t from adopting the VAR model over the SQR model is roughlysimilar in-and out-of-sample. It is also consistent with the evidence from returns and realizedvariances discussed earlier.

The improvement in t of the 3/2N model over the SQR model is less impressive but stillsubstantial. Recall that when judging the models performance using the MLISobjective func-tions in Table 1, the most impressive return-based performance of the 3/2N model obtained whenthe 1987 crash was included in the sample. Our evidence on the 3/2N model is interesting inlight of the ndings in Jones (2003) and At-Sahalia and Kimmel (2007). These papers bothestimate a CEV model using a bivariate time series of index returns and options. However, they

obtain very dierent results. Jones (2003) estimates a CEV parameter of 1.33 using a 1986-2000sample, while At-Sahalia and Kimmels (2007) estimate for the CEV parameter is 0.65. Ourresults suggest that one potential reason for these conicting results is the dierent samples usedin these studies: At-Sahalia and Kimmels sample is 1990-2003, and therefore does not includethe 1987 crash. However, it must be noted that Jones (2003) also estimates a CEV parameter of1.17 using a 1988-2000 sample. It may thus be the case that richer option data sets are neededin order to reliably estimate the model parameters.

When comparing the option-based NLSIS estimates in Table 4 with the return-based estimatesin Table 1, we nd that the is generally smaller, indicating a slower mean-reversion of variancewhen options are driving the parameters. The parameter is generally lower in Table 4, with theSQR model as the notable exception. The volatility of variance parameter is generally lower

in Table 4, the exception again being the SQR model. The volatility risk premium parameter is generally small but positive for all models as expected. Finally, the correlation coecient, ,is slightly smaller (in absolute value) when estimated using both options and returns.

Not surprisingly, when compared with the NLSIS-based in-sample IVRMSEs in Table 4, theMLIS-based errors for Wednesday call data in Table 3 are clearly considerably larger, except forthe 3/2 model where the dierence is negligible. In most cases, the IVRMSEs for Wednesday putsand Thursday calls in Table 3 are also larger than the corresponding out-of-sample NLSIS-basedIVRMSEs in Table 4.

Finally, the footnote to Table 4 reports the IVRMSE from two benchmark models. First,the simple Black-Scholes model where volatility is kept constant at the average implied volatility

across all options and across the entire sample period. The resulting IVRMSE is 4.95% forthe Wednesday call sample, 5.52% for the Wednesday put sample, and 5.22% for the Thursdaysample. Note that, not surprisingly, all the SV models in Table 4 outperform the simple Black-Scholes model by a wide margin both in and out of sample. The second benchmark is theso-called ad-hoc Black-Scholes model where the implied volatility for each option is set to theaverage implied volatility across options on the same weekday of the previous week. This ad-

11 See for instance Bates (2000), Pan (2002) and Eraker (2004). In recent work, using a very dierent empiricalsetup, Broadie, Chernov and Johannes (2007) nd large improvements in-sample when including Poisson jumps.

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hoc model has an IVRMSE of 2.99% for the Wednesday call sample, 3.52% for the Wednesdayput sample, and 3.06% for the Thursday sample. Thus, only the VAR model outperforms thisbenchmark in all three samples. This result may be surprising but note that the ad-hoc modelallows for the estimation of 9*52 = 468 parameters versus only ve parameters in the SV models.These results are not easily comparable with other recent studies as the weekly estimated ad-hoc

benchmark is not often compared with SV models estimated on a long sample. Heston and Nandi(2000) provide a similar benchmark, but their GARCH models are estimated on much shortersamples.

4.6 Decomposing the Option Valuation Errors

To provide more insight into the models performance, we decompose the in-sample option val-uation errors along several dimensions. The out-of-sample results yield very similar conclusionsregarding the models relative performance, and are thus omitted from the tables and gures.Option valuation models can fail for several reasons: they may imply a biased estimate of the

latent volatility variable, which leads to biases in the valuation of at-the-money options. Alter-natively, one volatility model may outperform another in the maturity dimension if it providesbetter estimates of the term structure of volatility, or in the moneyness dimension. We investi-gate all these model aspects, and we also investigate the models performance as a function ofthe level of market volatility, as measured by the VIX.

Figures 9 and 10 plot the performance of the models in valuing at-the-money options overtime. Figure 9 plots the weekly IVRMSE for at-the-money options, dened as options withmoneyness S=X, between 0.975 and 1.025. The 3/2 model and particularly the VAR modeldisplay a relatively low and stable IVRMSE across time. The IVRMSE plots for the SQR modeland the three models with nonlinear drifts exhibit many more spikes.

The MSE-based objective function can be viewed as a sum of error variance and bias squared.

Figure 10 therefore shows the weekly IV bias (average data IV less average model IV) for theat-the-money options. It is clear that some of the spikes in Figure 9 are driven at least partlyby spikes in the bias. This is particularly the case for the models with nonlinear drift. Inaddition to the high-frequency movements evident in Figure 10, it also appears that all modelsdisplay a somewhat persistent bias. This would indicate that volatility models with more exibleautocorrelation functions, as provided for example by the multiple volatility component modelsin Bates (2000), are needed.

In Table 5, we report the in-sample IVRMSE by moneyness, maturity, and volatility levelas measured by the VIX. Figure 11 depicts the IVRMSE results by moneyness, maturity, andvolatility categories, for the three models with linear variance drift, a = 0, in the left column,and the three models with nonlinear drift, a = 1, in the right column. The impressive overallperformance of the VAR model (denoted by x) clearly originates from good performance invirtually all moneyness, maturity and volatility categories. The benchmark SQR model (denotedby o) on the other hand generally performs worst or near worst in all moneyness, maturity andvolatility categories. The only exception is for the lowest VIX category, where the benchmarkSQR model performs relatively well.

In general, the dierences between models are larger within the class of models with lineardrift (the left column of Figure 11). The right column in Figure 11 indicates that the three modelswith nonlinear drift perform quite similarly across the moneyness and maturity dimension, and

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that in the volatility dimension the models only dier substantially for the highest VIX category.Comparing the two bottom panels, we see that the nonlinear variance drift improves the tacross the three diusion specications for the lowest VIX category, but it actually hurts theperformance of all three models for the highest VIX category.

The IVRMSE is generally increasing in moneyness for most models, which may be partly

driven by the fact that the average implied volatility is increasing in moneyness (see Table2). Table 5 indicates that the VAR model performs better than average in all six moneynesscategories. The VAR model performs best at pricing deep-in-the money calls. The 3/2N modelis also better than average for all categories, and is particularly good at valuing out-of-the-money calls. The SQR model performs below average in all moneyness categories, and it doesparticularly poorly for in-the-money options.

Note also that the IVRMSE tends to decrease with maturity for all models. This is onlypartially explained by the average implied volatility being larger for short-term options, becausethese dierences are small (see Table 2). Panel B of Table 5 indicates that the VAR modeloutperforms the other ve models for the most actively traded options (with maturity up to 60

days), and the VAR and 3/2N models are the best performers for maturities between 60 and 180days. For the longest maturities, the VARN model performs well. The VAR and the 3/2N modelsare better than average performers in all maturity categories. The SQR model performs worsethan average for all maturity categories and is the worst model in the four medium-maturitycategories.

The moneyness and maturity patterns clearly indicate that while non-ane volatility dynam-ics may lead to improved model performance, important biases remain. A potential solution toremedy these bias patterns is to include return jumps in the model specication. Negative return

jumps are likely to have most eect on the valuation of short-maturity and deep-in-the-moneycall options.

For the analysis by VIX level in panel C of Table 5 and the bottom panels in Figure 11, we

split the sample days into six categories sorted by the VIX level on each day of the sample. The3/2N model is better than average for all VIX categories, and the VAR model is better thanaverage for all except the lowest VIX levels. The SQR model performs well for the very lowestVIX level category, but performs poorly otherwise.

Figure 12 and Table 6 report the bias (average market price less average model price) inpercent across moneyness, maturity and volatility level categories, dened as in Table 5. Noterst and foremost from Table 6 that the overall bias is close to zero in all models, except for theSQR model where it is 1.06%, and for the 3/2 model, which has a bias of -0.90%.

The bias patterns in the top row of Figure 12 conrm a potentially important role for jumpsacross model specications. Panel A of Table 6 indicates that the SQR model underprices all

options, and all models underprice the deep in-the-money calls. All models except for the SQRmodel tend to overprice out-of-the money calls.Panel B of Table 6 reports the bias by maturity. The SQR model underprices all maturity

categories and the 3/2 model overprices all maturity categories. The remaining models tendto overprice short-term options and underprice long-term options but the biases are generallysmall. Interestingly, the middle row of Figure 12 highlights that the bias is generally smaller inmagnitude across maturity in the models with nonlinear drift (right column), suggesting thatsome of the IVRMSE patterns across maturity in Figure 11 originate from dierences in error

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variance across models. The nonlinear drift specication particularly helps in the SQR modelwhere the bias is large when the drift is linear.

Panel C of Table 6 reports the bias by volatility level. The bias tends to be increasing in VIXfor all models. The dierences in bias patterns between models across VIX categories clearlyexplain a signicant part of the IVRMSE VIX patterns in Figure 11. Recall that the bottom-left

panel in Figure 11 showed that the SQR model performs relatively well when VIX was at itslowest level. The bottom-left panel in Figure 12 reveals that this result is driven partly by asmaller bias. The bottom row in Figure 11 also showed that specifying a nonlinear variance driftimproves IVRMSE when volatility is low, but worsens IVRMSE when volatility is high. Thebottom row of Figure 12 gives at least a partial explanation: the negative bias observed at lowvolatility levels across diusion specications is worst when the drift is linear. The positive biasobserved at high volatility levels across diusion specications on the other hand is worst whenthe drift is nonlinear. We conclude that a more exible volatility specication may therefore beneeded.

4.7 Alternative Data and Filtering Techniques

The NLSIS estimation on option data leads to a number of interesting conclusions. The SQRmodel displays substantial biases and seems misspecied. While the option valuation errors andbias patterns for the other ve models suggest a role for alternative model specications, such asmultiple volatility components and return jumps, some of the models substantially outperformthe SQR model in most dimensions. This is particularly the case for the VAR model. Theseresults conrm our nding, obtained using realized volatility diagnostics and returns-based MLISestimation, that the SQR model is misspecied.

We now further investigate the robustness of these ndings with respect to the choice ofltering technique and the data used in ltering. While the PF used in the MLIS and NLSIS

estimation is computationally ecient and reliable, it is of interest to see if similar results obtainwith alternative ltering techniques. Moreover, while our results are largely consistent acrossthe MLIS and NLSIS ltering exercises that use daily returns, it is of interest to investigate ifsimilar results obtain when ltering the spot volatility from alternative data.

Table 7 reports on these robustness exercises. We consider two alternative data sources forltering: the daily realized variance data for 1996-2004 considered in Section 2.1, and the dailyVIX data from Section 2.3 for the same period. In Table 7, we rst value the Wednesday calloptions using the raw realized variance and (squared) VIX data as proxies for the spot variance.Subsequently, we lter the latent variance from these two series using the Kalman lter. We usethe simplest possible setup to lter from the VIX and RV series, ignoring returns altogether. Welimit our analysis to the SQR, VAR and 3/2 models, because the measurement equations relatingrealized volatility and VIX to the latent volatility are not known for models with nonlinear drift.Valuation results are obtained using the NLSIS-based parameter estimates from Table 4.

When ltering volatility from realized volatility, a sensible state-space representation for theSQR, VAR and 3/2 models is

RVt;t+1 = E[RVt;t+1jVt] + utVt+1 = Vt+ ( Vt) + Vbtwt+1

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Using the results in At-Sahalia and Kimmel (2007), this can be written as

RVt;t+1 = +

exp(=252)1

(=252)

(Vt ) + ut (30)

Vt+1 = Vt+ (

Vt) + Vbt wt+1

For the VIX, we employ the following state-space representation

V IXt;t+30 = +

exp(30=252)1

(30=252)

(Vt ) + ut (31)

Vt+1 = Vt+ ( Vt) + Vbtwt+1Table 7 presents the IVRMSEs, as well as the IVRMSE ratio between the VAR and 3/2

models and the SQR model. We also report the DM test for the VAR and 3/2 models againstthe SQR benchmark. The main conclusion is that the results conrm that the SQR model canbe substantially improved on by either of the two non-ane models. The IVRMSEs for the 3/2model are smaller than those for the VAR model, but in most cases the dierences are small.

In general, the IVRMSEs using the raw RV data are large compared to the MLIS- and NLSIS-based IVRMSEs in Tables 3 and 4. This is not surprising because these raw data contain moreoutliers than the variance data ltered from returns. When ltering the RV using the Kalmanlter setup in (30), the IVRMSEs are substantially lower than when using raw RV data. TheVIX data yield a lower IVRMSE than the realized volatility data and sometimes lower than thereturn-based ltering. This is also not surprising, because this implementation does not enforceconsistency with the daily return on the underlying index. The IVRMSEs for the VAR and 3/2models are similar, and both are substantially lower than the IVRMSE for the SQR model.

In summary, the results obtained using alternative data sources and ltering techniques con-

rm the misspecication of the SQR model, and indicate that non-ane models can oer sub-stantial improvements.

5 Summary and Conclusions

This paper provides an empirical comparison of the ane SQR model of Heston (1993) with arange of non-ane but equally parsimonious option valuation models. The main conclusion isthat non-ane specications can substantially improve upon the ane SQR model. This ndingis robust across a wide range of data sources used for volatility ltering, as well as across dierentestimation techniques. First, the misspecication of the SQR model is suggested by the stylized

facts characterizing realized volatility data. Second, an extensive estimation exercise on indexreturns conrms the presence of misspecication by analyzing measures of t. Third, non-anemodels lead to substantially smaller in-sample errors in option valuation when estimating usinga dataset consisting of long time series of cross sections of call option data. These results areconrmed when using the estimates out-of-sample by pricing options on dierent days and bypricing put options. Moreover, the model t obtained by valuing options using estimates fromreturns also conrms that the SQR model is misspecied. Fourth, option pricing errors are alsosmaller for non-ane models when using realized volatility and the VIX as a proxy for spot

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volatility, and when using the Kalman lter to obtain spot volatility from realized volatility andthe VIX.

We therefore conclude that while the option valuation literatures focus on ane models is wellmotivated, because the resulting closed-form solutions are extremely convenient, this analyticalconvenience comes at a price, and non-ane models need to be studied more extensively. Our

empirical results provide some important guidance for the specication of non-ane models. TheVAR model consistently performs very well. It provides the best t for two of the three returnsamples, and the second best t using the third sample. When estimating the models usingoptions data, it is the best performing model in- and out-of-sample. When using the returns-based parameter estimates to value options, the ranking of the ve non-ane models dependson the data used.

At the methodological level, this paper uses a method to estimate continuous-time option val-uation models that has not yet been applied to this particular problem. We use particle ltering,which is rather exible and straightforward to implement. It can easily be used to investigateoption data and underlying equity returns jointly for a wide range of objective functions as well

as for a wide range of models. In our opinion, because of its simplicity and numerical eciency,this method is attractive compared to other methods that have been used in this literature.Our empirical results suggest a number of important extensions. First, while non-ane mod-

els substantially outperform ane models, the biases of the ane model can also be addressed byalternative model specications, such as multiple volatility components, and/or Poisson or Levy

jumps in returns and volatility.12 This study analyzes non-ane specications without consider-ing these alternatives, but ultimately it will be of interest to investigate the relative advantages ofnon-ane specications, multiple volatility components, and jump processes. Second, the ndingfrom the returns data that the 3/2N model is particularly useful in a sample that includes the1987 crash, indicates that another interesting avenue for future work is to focus more explicitly onthe CEV model, and particularly to investigate its performance when periodically re-estimating

the CEV parameter. Third, the analysis in Section 4.7 provides some interesting insights andmay be worth extending. In particular, model estimates obtained by ltering volatility usingreturns and the cross-section of options could be compared with model estimates obtained byltering using returns and either the VIX or the realized volatility data or both.

6 Appendix

In this appendix, we rst study the properties of Monte Carlo based option prices. We compareMonte Carlo prices for the Heston (1993) SQR model with prices computed via Fourier inversionof the conditional characteristic function. We then study the nite sample properties of the MLIS

estimator, comparing it with other available estimators in the literature.

6.1 Option Price Computation by Simulation

To assess the accuracy of our Monte Carlo setup, we compute a set of analytical option pricesusing Hestons (1993) model for various strike prices, moneyness and maturities. We then com-pute Monte Carlo option prices for the same options. Using the numerical example in Heston

12 See Carr and Wu (2004) and Huang and Wu (2004) for option valuation studies using Levy processes.

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(1993), the risk-neutral variance process is parameterized as follows

dV = 2(:01 V)dt + :2p

V dw (32)

The correlation is set to

0:5, and the risk-free rate is set to zero.

Figure A.1 reports the results. The Monte Carlo prices, denoted by +, oer a very goodapproximation to the analytical prices, denoted by solid lines, in all cases. The gure contains twomaturities: one month (left column) and three months (right column). The gure considers threespot variance levels: V0= :005, which is half the unconditional variance (top row), V0= :01whichis equal to the unconditional variance (middle row), and V0= :02which is twice the unconditionalvariance (bottom row).

6.2 Finite Sample Performance of MLIS

We assess the nite sample performance of the relatively new MLIS estimation procedure basedon the particle lter. Following Bates (2006), who introduces a new approximate ML estima-tion procedure, we add the MLIS estimator to the large-scale Monte Carlo study in Andersen,Chung and Srensen (1999), henceforth ACS. ACS use the following simple SV model as a datagenerating process

ln(St+1) = ln(St) +p

Vtzt+1 (33)

ln (Vt+1) =!+ ln (Vt) + wt+1 (34)

wherezt+1 andwt+1 are uncorrelated. They consider a large number of estimators including theQML method from Harvey, Ruiz and Shephard (1994), the GMM from Andersen and Srensen(1996), MCMC from Jacquier, Polson and Rossi (1994), as well as the EMM implementation

suggested by ACS themselves. Table A.1 reproduces the results from ACS and includes theapproximate maximum likelihood (AML) technique from Bates (2006). We of course also includethe MLIS estimation procedure from Pitt (2002) which is used in this paper. The variousestimators are compared to the benchmark (but of course unrealistic) case where the spot varianceis observed and standard ML estimation is straightforward. We report results for the sample sizeT = 2; 000 which is relevant for our subsequent empirical study.

Table A.1 reports parameter bias and root mean squared error (RMSE) for each estimatorusing 500 Monte Carlo replications. The MCMC estimator has the lowest absolute bias for the! parameter followed by the AML and the MLIS. The MLIS, MCMC and AML estimators havethe lowest absolute biases for. The AML, EMM and MCMC estimators have the lowest biasesfor the parameter followed by the MLIS. The MCMC and MLIS have the lowest RMSE for

!; AML, MCMC and MLIS have the lowest RMSE for , and MCMC has the lowest RMSEfor the parameter followed by AML and MLIS. Overall, the MLIS estimator seems to performwell. The fact that it is also easily implementable across a wide range of models and objectivefunctions makes it a very suitable estimation method in our large-scale empirical study.

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